Ring-shaped molecules are commonly found both in nature and in man-made compounds. In a ring-shaped molecule, the atoms forming the molecule are interconnected to each other to form a loop or ring, such as shown in FIG. 1. Viewed in two-dimensions, the distances between atoms, known as the bond lengths, and the angles formed at an atom and defined by its bonds with two other atoms, known as the bond angles, are well-known and can be readily determined experimentally. However, viewed in three-dimensions, these atoms may bond in a variety of rotations, identified by torsion angles. Accordingly, a ring-shaped molecule may have many shapes in three-dimensions. The term "shape" of a molecule signifies the arrangement, configuration or location of atoms forming the molecule with respect to each other in three-dimensions. In this field, shape is typically synonomous with "conformation." A "conformer" is a molecule with a particular shape. The shapes of ring-shaped structures are generally given descriptive names in the field of chemistry, such as a "boat", a "chair", a "sofa" or an "envelope." There are multitudes of other shapes.
Molecules including one or more ring-shaped structures are particularly common in biological systems. Most familiar, perhaps, are the carbohydrates in general and the simple sugars (monosaccharides) in particular. For example, the pyranose sugars have six-membered rings comprised of five carbon atoms and a single oxygen atom. These rings occur in the common monosaccharides (glucose, fructose, galactose and mannose) as well as their monosaccharide derivatives such as acetals (e.g., the glucosides), ethers (e.g., 2,3,4,6-tetramethylgluco-pyranose), and acids (e.g. glucuronic acid). The furanoses, a group of common sugars (e.g., arabinose or ribose) in which four carbon atoms and a single oxygen atom form a five-membered ring. Side groups (substituents) which may be attached to the atoms in these rings are essentially unlimited in number and may constitute the better part of the mass of the molecule (e.g., the gangliosides and nucleosides). More generally, ring-shaped structures including anywhere from three to dozens of atoms (e.g., porphyrins and other metalocycles) may be found in natural systems with a wide variety of side groups attached to The ring-shaped structures. In addition, the rings may contain heteroatoms, partial double, double or triple bonds (e.g. Phe, Tyr, dopamine, steroids, vitamins).
These ring-shaped structures are typically flexible and can shift between various three-dimensional shapes while in solution. The so-called "chair" and "boat" shapes of six-membered rings are the most famous examples in which rotation around the bonds forming the ring can cause the atoms to change their spatial orientation with respect to one another while still being constrained by the bonds of the ring. The flexibility of rings will, of course, decrease with decreasing ring size (e.g., cyclopropane) and with the number of double bonds, partial double bonds or triple bonds between the ring atoms (e.g., benzene and other cycloalkenes). In addition, the side groups on a ring may cause the ring preferentially or exclusively to assume a particular shape due to forces such as steric hindrance or electrostatic attraction or repulsion among themselves. Nonetheless, many ring structures of biological interest retain a great deal of flexibility.
Many biologically significant molecules have a structure which consists of a chain or polymer of many repeating subunits which include a ring-shaped structure. The most common examples are, again, the carbohydrates in general and the polysaccharides in particular. Cellulose and amylose, for example, each consist of a polymer of glucose rings joined by 1-4 glucoside linkages. More complex polymers with repeating disaccharide or oligosaccharide subunits and with a wide variety of side groups with varying complexity (e.g., glycoproteins, gangliosides, glycosaminoglycans and mucopolysaccharides) are also quite common. In addition, non-carbohydrates with repeating ring structures are also among the most important biological molecules known. For example, the backbone of the ribonucleic acids is a polymer of ribose rings which have purine or pyrimidine bases as side groups and which are linked by phosphate esters.
The three-dimensional shape of biologically active molecules is often as important or more important than their molecular composition. The active sites of enzymes and the binding sites of cell receptors and antigens in these molecules typically have a three-dimensional shape which is complementary to the shape of their substrates, ligands and antigens. Therefore, the function and selectivity of a portion of a given molecule is highly dependent upon its shape. This shape may be dependent on a ring-shaped structure in either the polymer backbone of the molecule or a side group of the molecule. The ability to understand and estimate these shapes is, therefore, often crucial to understanding the biological activity of many molecules. In addition, the rapidly developing field of rational drug design is based in large part upon the development of new molecules which mimic the shape of natural molecules, or portions thereof, but which have different reactivity. Competitive inhibitors of an enzyme, for example, typically have a similar shape and similar charge characteristics to the natural substrate but differ in molecular composition such that they may occupy the active site of an enzyme but are not subject to the chemical reaction which the enzyme normally catalyzes. A better understanding of the shape of molecules including one or more ring structures, therefore, is valuable both in assessing the three-dimensional structure of a natural molecule which is sought to be mimicked and in predicting the utility of a proposed mimic.
The carbohydrate chains of glycoproteins and proteoglycans are just now being recognized as being of immense biological importance. The importance of oligosaccharides is discussed in "Biological Roles of Oligosaccharides" by Ajit Varki in Glycobiology, Vol. 3, No. 2, pp. 97-130 (1993). These polymers of ring structures may be found bound to membrane proteins, bound to the extracellular matrix, as integral components of basement membranes, or as soluble components of the extracellular matrix. As they may have functions in cell recognition, cell adhesion, cell migration, antigen recognition, cytokine presentation or activation, immobilization of growth factors and enzymes, and clearance from circulation, these carbohydrates are now the subject of intense scrutiny and knowledge of their three-dimensional structure is greatly needed. Among those of greatest interest are glycosaminoglycans such as chondroitin-6-sulfate, dermatan sulfate, heparin, heparon sulfate, keratan sulfate, and hyaluronic acid, all of which are polymers of substituted pyranose disaccharides.
The importance of the shape of a ring-shaped molecule or of a molecule including a ring-shaped structure is not limited to biological environments. Such molecules also occur in synthetic polymers (e.g., cellophane and celluloid), paper (e.g., cellulose) and food (e.g., glucose). With respect to synthetic polymers, the properties of synthetic polymers depend on the interaction between each of the monomeric units of which it is comprised. Estimating the shape of one polymer molecule allows one to study the way a multiplacity of these polymers assemble in three dimensions and to determine bulk properties. Knowledge from such study also allows polymers to be designed to have specific properties. A better understanding of the shapes of such molecules, and the ability to predict the shape of newly proposed materials, such as synthetic polymers, remain critical needs.
There are some computer systems which are used to estimate the shape of a molecule which contains ring-shaped structures, but these systems all assume that each of the ring-shaped structures within the molecule takes some predetermined shape. An expert typically estimates this shape or configuration. Given the assumption by the expert that the ring-shaped structure has a particular shape or configuration, the shapes of chains of this structure are then examined using conventional techniques. Such methods are described, for example, by Grootenhuis, P. D. J., et al., in "Carbohydrates and Drug Discovery--the Role of Computer Simulation," Tibtech, January 1994 (Vol. 12), pp. 9-14; French, A. D., et al., in Computer Moleling of Carbohydrate Molecules, eds. French, A. D. and Brady, J. W. (Am. Chem. Soc., Washington, D.C.) pp. 1-19; and by Homans, S. W., et al., in "Oligosaccharides and Recognition--A `Shape` Problem Probed By N.M.R. and Molecular Modeling," Biochemical Society Transactions, Carbohydrates, Shapes and Biological Recognition, Vol. 21, 1993, pp. 449-452.